Determinants and their applications in mathematical physics
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| Format: | Buch |
| Sprache: | German |
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New York [u.a.]
Springer
1999
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| Schriftenreihe: | Applied mathematical sciences
134 |
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| 100 | 1 | |a Vein, Robert |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Determinants and their applications in mathematical physics |c Robert Vein ; Paul Dale |
| 264 | 1 | |a New York [u.a.] |b Springer |c 1999 | |
| 300 | |a XIV, 376 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Applied mathematical sciences |v 134 | |
| 500 | |a Literaturverz. S. 343 - 372 | ||
| 650 | 4 | |a Determinantes | |
| 650 | 4 | |a Física matemática | |
| 650 | 4 | |a Mathematische Physik | |
| 650 | 4 | |a Determinants | |
| 650 | 4 | |a Mathematical physics | |
| 650 | 0 | 7 | |a Determinantentheorie |0 (DE-588)4285603-6 |2 gnd |9 rswk-swf |
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| 700 | 1 | |a Dale, Paul |e Verfasser |4 aut | |
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Datensatz im Suchindex
| _version_ | 1819594458502529024 |
|---|---|
| adam_text | Contents
Preface v
1 Determinants, First Minors, and Cofactors 1
1.1 Grassmann Exterior Algebra 1
1.2 Determinants 1
1.3 First Minors and Cofactors 3
1.4 The Product of Two Determinants — 1 5
2 A Summary of Basic Determinant Theory 7
2.1 Introduction 7
2.2 Row and Column Vectors 7
2.3 Elementary Formulas 8
2.3.1 Basic Properties 8
2.3.2 Matrix Type Products Related to Row and
Column Operations 10
2.3.3 First Minors and Cofactors; Row and Column
Expansions 12
2.3.4 Alien Cofactors; The Sum Formula 12
2.3.5 Cramer s Formula 13
2.3.6 The Cofactors of a Zero Determinant 15
2.3.7 The Derivative of a Determinant 15
3 Intermediate Determinant Theory 16
3.1 Cyclic Dislocations and Generalizations 16
3.2 Second and Higher Minors and Cofactors 18
3.2.1 Rejecter and Retainer Minors 18
3.2.2 Second and Higher Cofactors 19
3.2.3 The Expansion of Cofactors in Terms of Higher
Cofactors 20
3.2.4 Alien Second and Higher Cofactors; Sum
Formulas 22
3.2.5 Scaled Cofactors 23
3.3 The Laplace Expansion 25
3.3.1 A Grassmann Proof 25
3.3.2 A Classical Proof 27
3.3.3 Determinants Containing Blocks of Zero Elements . 30
3.3.4 The Laplace Sum Formula 32
3.3.5 The Product of Two Determinants — 2 33
3.4 Double Sum Relations for Scaled Cofactors 34
3.5 The Adjoint Determinant 36
3.5.1 Definition 36
3.5.2 The Cauchy Identity 36
3.5.3 An Identity Involving a Hybrid Determinant ... 37
3.6 The Jacobi Identity and Variants 38
3.6.1 The Jacobi Identity — 1 38
3.6.2 The Jacobi Identity — 2 41
3.6.3 Variants 43
3.7 Bordered Determinants 46
3.7.1 Basic Formulas; The Cauchy Expansion 46
3.7.2 A Determinant with Double Borders 49
4 Particular Determinants 51
4.1 Alternants 51
4.1.1 Introduction 51
4.1.2 Vandermondians 52
4.1.3 Cofactors of the Vandermondian 54
4.1.4 A Hybrid Determinant 55
4.1.5 The Cauchy Double Alternant 57
4.1.6 A Determinant Related to a Vandermondian ... 59
4.1.7 A Generalized Vandermondian 60
4.1.8 Simple Vandermondian Identities 60
4.1.9 Further Vandermondian Identities 63
4.2 Symmetric Determinants 64
4.3 Skew Symmetric Determinants 65
4.3.1 Introduction 65
4.3.2 Preparatory Lemmas 69
4.3.3 Pfaffians 73
4.4 Circulants 79
4.4.1 Definition and Notation 79
4.4.2 Factors 79
4.4.3 The Generalized Hyperbolic Functions 81
4.5 Centrosymmetric Determinants 85
4.5.1 Definition and Factorization 85
4.5.2 Symmetric Toeplitz Determinants 87
4.5.3 Skew Centrosymmetric Determinants 90
4.6 Hessenbergians 90
4.6.1 Definition and Recurrence Relation 90
4.6.2 A Reciprocal Power Series 92
4.6.3 A Hessenberg Appell Characteristic Polynomial . 94
4.7 Wronskians 97
4.7.1 Introduction 97
4.7.2 The Derivatives of a Wronskian 99
4.7.3 The Derivative of a Cofactor 100
4.7.4 An Arbitrary Determinant 102
4.7.5 Adjunct Functions 102
4.7.6 Two Way Wronskians 103
4.8 Hankelians 1 104
4.8.1 Definition and the 4 m Notation 104
4.8.2 Hankelians Whose Elements are Differences .... 106
4.8.3 Two Kinds of Homogeneity 108
4.8.4 The Sum Formula 108
4.8.5 Turanians 109
4.8.6 Partial Derivatives with Respect to tpm HI
4.8.7 Double Sum Relations 112
4.9 Hankelians 2 115
4.9.1 The Derivatives of Hankelians with Appell
Elements 115
4.9.2 The Derivatives of Turanians with Appell and
Other Elements 119
4.9.3 Determinants with Simple Derivatives of All
Orders 122
4.10 Henkelians 3 123
4.10.1 The Generalized Hilbert Determinant 123
4.10.2 Three Formulas of the Rodrigues Type 127
4.10.3 Bordered Yamazaki Hori Determinants — 1 .... 129
4.10.4 A Particular Case of the Yamazaki Hori
Determinant 135
4.11 Hankelians 4 137
4.11.1 u Numbers 137
4.11.2 Some Determinants with Determinantal Factors . 138
4.11.3 Some Determinants with Binomial and Factorial
Elements 142
4.11.4 A Nonlinear Differential Equation 147
4.12 Hankelians 5 153
4.12.1 Orthogonal Polynomials 153
4.12.2 The Generalized Geometric Series and Eulerian
Polynomials 157
4.12.3 A Further Generalization of the Geometric Series . 162
4.13 Hankelians 6 165
4.13.1 Two Matrix Identities and Their Corollaries .... 165
4.13.2 The Factors of a Particular Symmetric Toeplitz
Determinant 168
4.14 Casoratians — A Brief Note 169
5 Further Determinant Theory 170
5.1 Determinants Which Represent Particular Polynomials . . 170
5.1.1 Appell Polynomial 170
5.1.2 The Generalized Geometric Series and Eulerian
Polynomials 172
5.1.3 Orthogonal Polynomials 174
5.2 The Generalized Cusick Identities 178
5.2.1 Three Determinants 178
5.2.2 Four Lemmas 180
5.2.3 Proof of the Principal Theorem 183
5.2.4 Three Further Theorems 184
5.3 The Matsuno Identities 187
5.3.1 A General Identity 187
5.3.2 Particular Identities 189
5.4 The Cofactors of the Matsuno Determinant 192
5.4.1 Introduction 192
5.4.2 First Cofactors 193
5.4.3 First and Second Cofactors 194
5.4.4 Third and Fourth Cofactors 195
5.4.5 Three Further Identities 198
5.5 Determinants Associated with a Continued Fraction . . . 201
5.5.1 Continuants and the Recurrence Relation 201
5.5.2 Polynomials and Power Series 203
5.5.3 Further Determinantal Formulas 209
5.6 Distinct Matrices with Nondistinct Determinants 211
5.6.1 Introduction 211
5.6.2 Determinants with Binomial Elements 212
5.6.3 Determinants with Stirling Elements 217
5.7 The One Variable Hirota Operator 221
5.7.1 Definition and Taylor Relations 221
5.7.2 A Determinantal Identity 222
5.8 Some Applications of Algebraic Computing 226
5.8.1 Introduction 226
5.8.2 Hankel Determinants with Hessenberg Elements . 227
5.8.3 Hankel Determinants with Hankel Elements .... 229
5.8.4 Hankel Determinants with Symmetric Toeplitz
Elements 231
5.8.5 Hessenberg Determinants with Prime Elements . . 232
5.8.6 Bordered Yamazaki Hori Determinants — 2 .... 232
5.8.7 Determinantal Identities Related to Matrix
Identities 233
6 Applications of Determinants in Mathematical Physics 235
6.1 Introduction 235
6.2 Brief Historical Notes 236
6.2.1 The Dale Equation 236
6.2.2 The Kay Moses Equation 237
6.2.3 The Toda Equations 237
6.2.4 The Matsukidaira Satsuma Equations 239
6.2.5 The Korteweg de Vries Equation 239
6.2.6 The Kadomtsev Petviashvili Equation 240
6.2.7 The Benjamin Ono Equation 241
6.2.8 The Einstein and Ernst Equations 241
6.2.9 The Relativistic Toda Equation 245
6.3 The Dale Equation 246
6.4 The Kay Moses Equation 249
6.5 The Toda Equations 252
6.5.1 The First Order Toda Equation 252
6.5.2 The Second Order Toda Equations 254
6.5.3 The Milne Thomson Equation 256
6.6 The Matsukidaira Satsuma Equations 258
6.6.1 A System With One Continuous and One
Discrete Variable 258
6.6.2 A System With Two Continuous and Two
Discrete Variables 261
6.7 The Korteweg de Vries Equation 263
6.7.1 Introduction 263
6.7.2 The First Form of Solution 264
6.7.3 The First Form of Solution, Second Proof 268
6.7.4 The Wronskian Solution 271
6.7.5 Direct Verification of the Wronskian Solution . . . 273
6.8 The Kadomtsev Petviashvili Equation 277
6.8.1 The Non Wronskian Solution 277
6.8.2 The Wronskian Solution 280
6.9 The Benjamin Ono Equation 281
6.9.1 Introduction 281
6.9.2 Three Determinants 282
6.9.3 Proof of the Main Theorem 285
6.10 The Einstein and Ernst Equations 287
6.10.1 Introduction 287
6.10.2 Preparatory Lemmas 287
6.10.3 The Intermediate Solutions 292
6.10.4 Preparatory Theorems 295
6.10.5 Physically Significant Solutions 299
6.10.6 The Ernst Equation 302
6.11 The Relativistic Toda Equation — A Brief Note 302
A Appendix 304
A.I Miscellaneous Functions 304
A.2 Permutations 307
A.3 Multiple Sum Identities 311
A.4 Appell Polynomials 314
A.5 Orthogonal Polynomials 321
A.6 The Generalized Geometric Series and Eulerian
Polynomials 323
A.7 Symmetric Polynomials 326
A.8 Differences 328
A.9 The Euler and Modified Euler Theorems on Homogeneous
Functions 330
A.10 Formulas Related to the Function (x + y/l + x2)2n .... 332
A.11 Solutions of a Pair of Coupled Equations 335
A.12 Backlund Transformations 337
A. 13 Muir and Metzler, A Treatise on the Theory of
Determinants 341
Bibliography 343
Index 373
|
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| bvnumber | BV012409094 |
| callnumber-first | Q - Science |
| callnumber-label | QA1 QC20 |
| callnumber-raw | QA1 QC20.7.D45.A647 vol. 134 |
| callnumber-search | QA1 QC20.7.D45.A647 vol. 134 |
| callnumber-sort | QA 11 _Q C20 17. D45 A647 VOL 3134 |
| callnumber-subject | QA - Mathematics |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 510 - Mathematics 530 - Physics |
| dewey-raw | 510 530.15 |
| dewey-search | 510 530.15 |
| dewey-sort | 3510 |
| dewey-tens | 510 - Mathematics 530 - Physics |
| discipline | Physik Mathematik |
| format | Book |
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| id | DE-604.BV012409094 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-23T15:02:05Z |
| institution | BVB |
| isbn | 0387985581 |
| language | German |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008418616 |
| oclc_num | 318410594 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-703 DE-29T DE-20 DE-91G DE-BY-TUM DE-521 DE-634 DE-83 DE-11 DE-19 DE-BY-UBM |
| owner_facet | DE-355 DE-BY-UBR DE-703 DE-29T DE-20 DE-91G DE-BY-TUM DE-521 DE-634 DE-83 DE-11 DE-19 DE-BY-UBM |
| physical | XIV, 376 S. |
| publishDate | 1999 |
| publishDateSearch | 1999 |
| publishDateSort | 1999 |
| publisher | Springer |
| record_format | marc |
| series | Applied mathematical sciences |
| series2 | Applied mathematical sciences |
| spellingShingle | Vein, Robert Dale, Paul Determinants and their applications in mathematical physics Applied mathematical sciences Determinantes Física matemática Mathematische Physik Determinants Mathematical physics Determinantentheorie (DE-588)4285603-6 gnd Determinante (DE-588)4138983-9 gnd Mathematische Physik (DE-588)4037952-8 gnd |
| subject_GND | (DE-588)4285603-6 (DE-588)4138983-9 (DE-588)4037952-8 |
| title | Determinants and their applications in mathematical physics |
| title_auth | Determinants and their applications in mathematical physics |
| title_exact_search | Determinants and their applications in mathematical physics |
| title_full | Determinants and their applications in mathematical physics Robert Vein ; Paul Dale |
| title_fullStr | Determinants and their applications in mathematical physics Robert Vein ; Paul Dale |
| title_full_unstemmed | Determinants and their applications in mathematical physics Robert Vein ; Paul Dale |
| title_short | Determinants and their applications in mathematical physics |
| title_sort | determinants and their applications in mathematical physics |
| topic | Determinantes Física matemática Mathematische Physik Determinants Mathematical physics Determinantentheorie (DE-588)4285603-6 gnd Determinante (DE-588)4138983-9 gnd Mathematische Physik (DE-588)4037952-8 gnd |
| topic_facet | Determinantes Física matemática Mathematische Physik Determinants Mathematical physics Determinantentheorie Determinante |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008418616&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000005274 |
| work_keys_str_mv | AT veinrobert determinantsandtheirapplicationsinmathematicalphysics AT dalepaul determinantsandtheirapplicationsinmathematicalphysics |